Revista Matemática Complutense

, Volume 28, Issue 3, pp 549–597 | Cite as

The convenient setting for Denjoy–Carleman differentiable mappings of Beurling and Roumieu type

Article

Abstract

We prove in a uniform way that all Denjoy–Carleman differentiable function classes of Beurling type \(C^{(M)}\) and of Roumieu type \(C^{\{M\}}\), admit a convenient setting if the weight sequence \(M=(M_k)\) is log-convex and of moderate growth: For \(\mathcal C\) denoting either \(C^{(M)}\) or \(C^{\{M\}}\), the category of \(\mathcal C\)-mappings is cartesian closed in the sense that \(\mathcal C(E,\mathcal C(F,G))\cong \mathcal C(E\times F, G)\) for convenient vector spaces. Applications to manifolds of mappings are given: The group of \(\mathcal C\)-diffeomorphisms is a regular \(\mathcal C\)-Lie group if \(\mathcal C \supseteq C^\omega \), but not better.

Keywords

Convenient setting Denjoy–Carleman classes of Roumieu and Beurling type Quasianalytic and non-quasianalytic mappings of moderate growth Whitney jets on Banach spaces 

Mathematics Subject Classification

26E10 46A17 46E50 58B10 58B25 58C25 58D05 58D15 

References

  1. 1.
    Bierstone, E., Milman, P.D.: Resolution of singularities in Denjoy–Carleman classes. Selecta Math. (N.S.) 10(1), 1–28 (2004)Google Scholar
  2. 2.
    Bruna, J.: On inverse-closed algebras of infinitely differentiable functions, Studia Math. 69(1), 59–68 (1980/81)Google Scholar
  3. 3.
    Carleman, T.: Les fonctions quasi-analytiques. Collection Borel. Gauthier-Villars, Paris (1926)Google Scholar
  4. 4.
    Chaumat, J., Chollet, A.-M.: Surjectivité de l’application restriction à un compact dans des classes de fonctions ultradifférentiables. Math. Ann. 298(1), 7–40 (1994)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chaumat, J., Chollet, A.-M.: Propriétés de l’intersection des classes de Gevrey et de certaines autres classes. Bull. Sci. Math. 122(6), 455–485 (1998)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Denjoy, A.: Sur les fonctions quasi-analytiques de variable réelle. C. R. Acad. Sci. Paris 173, 1320–1322 (1921)Google Scholar
  7. 7.
    Dyn’kin, E.M.: Pseudoanalytic extension of smooth functions. The uniform scale. Am. Math. Soc. Transl. Ser. 2 2115, 33–58 (1976)Google Scholar
  8. 8.
    Faà di Bruno, C.F.: Note sur une nouvelle formule du calcul différentielle. Q. J. Math. 1, 359–360 (1855)Google Scholar
  9. 9.
    Frölicher, A.: Catégories cartésiennement fermées engendrées par des monoïdes, Cahiers Topologie Géom. Différentielle 21(4), 367–375 (1980). Third Colloquium on Categories (Amiens, 1980), Part IGoogle Scholar
  10. 10.
    Frölicher, A.: Applications lisses entre espaces et variétés de Fréchet. C. R. Acad. Sci. Paris Sér. I Math. 293(2), 125–127 (1981)Google Scholar
  11. 11.
    Frölicher, A., Kriegl, A.: Linear Spaces and Differentiation Theory. Pure and applied mathematics (New York). John Wiley & Sons Ltd., Chichester (1988)Google Scholar
  12. 12.
    Grabowski, J.: Free subgroups of diffeomorphism groups. Fund. Math. 131(2), 103–121 (1988)MathSciNetGoogle Scholar
  13. 13.
    Grauert, H.: On Levi’s problem and the imbedding of real-analytic manifolds. Ann. Math. (2) 68, 460–472 (1958)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hörmander, L.: The analysis of linear partial differential operators. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Distribution theory and Fourier analysis, vol. 256. Springer, Berlin (1983)Google Scholar
  15. 15.
    Komatsu, H.: Ultradistributions. I. Structure theorems and a characterization. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20, 25–105 (1973)MathSciNetGoogle Scholar
  16. 16.
    Komatsu, H.: An analogue of the Cauchy–Kowalevsky theorem for ultradifferentiable functions and a division theorem for ultradistributions as its dual. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 26(2), 239–254 (1979)MathSciNetGoogle Scholar
  17. 17.
    Komatsu, H.: The implicit function theorem for ultradifferentiable mappings. Proc. Jpn. Acad. Ser. A Math. Sci. 55(3), 69–72 (1979)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Komatsu, H.: Ultradifferentiability of solutions of ordinary differential equations. Proc. Jpn. Acad. Ser. A Math. Sci. 56(4), 137–142 (1980)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Koosis, P.: The Logarithmic Integral. I. Cambridge studies in advanced mathematics, vol. 12. Cambridge University Press, Cambridge (1998) (Corrected reprint of the 1988 original)Google Scholar
  20. 20.
    Kriegl, A.: Die richtigen Räume für Analysis im Unendlich-Dimensionalen. Monatsh. Math. 94(2), 109–124 (1982)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kriegl, A.: Eine kartesisch abgeschlossene Kategorie glatter Abbildungen zwischen beliebigen lokalkonvexen Vektorräumen. Monatsh. Math. 95(4), 287–309 (1983)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Kriegl, A., Michor, P.W.: The convenient setting for real analytic mappings. Acta Math. 165(1–2), 105–159 (1990)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Kriegl, A., Michor, P.W.: The convenient setting of global analysis, Mathematical Surveys and Monographs, vol. 53. American Mathematical Society, Providence (1997). http://www.ams.org/online_bks/surv53/
  24. 24.
    Kriegl, A., Michor, P.W.: Regular infinite-dimensional Lie groups. J. Lie Theory 7(1), 61–99 (1997)MathSciNetGoogle Scholar
  25. 25.
    Kriegl, A., Michor, P.W., Rainer, A.: The convenient setting for non-quasianalytic Denjoy–Carleman differentiable mappings. J. Funct. Anal. 256, 3510–3544 (2009)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kriegl, A., Michor, P.W., Rainer, A.: The convenient setting for quasianalytic Denjoy–Carleman differentiable mappings. J. Funct. Anal. 261, 1799–1834 (2011)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Kriegl, A., Michor, P.W., Rainer, A.: Denjoy–Carleman differentiable perturbation of polynomials and unbounded operators. Integral Equ. Oper. Theory 71(3), 407–416 (2011)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Langenbruch, M.: A general approximation theorem of Whitney type. RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 97(2), 287–303 (2003)MathSciNetGoogle Scholar
  29. 29.
    Neus, H.: Über die Regularitätsbegriffe induktiver lokalkonvexer Sequenzen. Manuscripta Math. 25(2), 135–145 (1978)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Rainer, A.: Perturbation theory for normal operators. Trans. Am. Math. Soc. 365(10), 5545–5577 (2013)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Rainer, A., Schindl, G.: Composition in ultradifferentiable classes. Studia Math. 224(2), 97–131 (2014)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Rainer, A., Schindl, G.: Equivalence of stability properties for ultradifferentiable classes. RACSAM. doi:10.1007/s13398-014-0216-0
  33. 33.
    Retakh, V.S.: Subspaces of a countable inductive limit. Sov. Math. Dokl. 11, 1384–1386 (1970)Google Scholar
  34. 34.
    Roumieu, C.: Ultra-distributions définies sur \(R^{n}\) et sur certaines classes de variétés différentiables. J. Anal. Math. 10, 153–192 (1962/1963)Google Scholar
  35. 35.
    Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill Book Co., New York (1987)Google Scholar
  36. 36.
    Schindl, G.: Spaces of smooth functions of Denjoy-Carleman-type. Diploma Thesis. http://othes.univie.ac.at/7715/1/2009-11-18_0304518 (2009)
  37. 37.
    Schmets, J., Valdivia, M.: On nuclear maps between spaces of ultradifferentiable jets of Roumieu type. RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 97(2), 315–324 (2003)MathSciNetGoogle Scholar
  38. 38.
    Thilliez, V.: On quasianalytic local rings. Expo. Math. 26(1), 1–23 (2008)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Yamanaka, T.: Inverse map theorem in the ultra-\(F\)-differentiable class. Proc. Jpn. Acad. Ser. A Math. Sci. 65(7), 199–202 (1989)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Yamanaka, T.: On ODEs in the ultradifferentiable class. Nonlinear Anal. 17(7), 599–611 (1991)MathSciNetCrossRefGoogle Scholar

Copyright information

© Universidad Complutense de Madrid 2015

Authors and Affiliations

  • Andreas Kriegl
    • 1
  • Peter W. Michor
    • 1
  • Armin Rainer
    • 1
  1. 1.Fakultät für MathematikUniversität WienWienAustria

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